Antenna processing method for potentially non-centered cyclostationary signals

ABSTRACT

An antenna processing method for centered or potentially non-centered cyclostationary signals, comprises at least one step in which one or more nth order estimators are obtained from r-order statistics, with r=1 to n−1, and for one or more values of r, it comprises a step for the correction of the estimator by means of r-order detected cyclic frequencies. The method can be applied to the separation of the emitter sources of the signals received by using the estimator or estimators.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to an antenna processing method for potentially non-centered or centered cyclostationary sources.

It can be applied for example to CPFSK sources with integer modulation index.

The invention relates, for example, to a method for the separation of potentially non-centered, cyclostationary signals received by a receiver of a communications system comprising several sources or emitters. The term “cyclostationary” also designates the particular case of stationary signals.

It can also be applied to the angular localization or goniometry of potentially non-centered cyclostationary sources.

The invention can be applied especially in radiocommunications, space telecommunications or passive listening to these links, in frequencies ranging from the VLF (Very Low Frequency) to the EHF (Extremely High Frequency).

In the present description, the term “blind separation” designates the separation of emitters with no knowledge whatsoever of the signals sent, the term “centered signal” refers to a signal without any continuous component that verifies E[x(t)]=0, and the term “non-stationary signal” refers to a signal whose statistics are time-dependent.

2. Description of the Prior Art

In many contexts of application, the reception of signals of interest for the receiver is very often disturbed by the presence of other signals (or sources) known as parasites, which may correspond either to delayed versions of the signals of interest (through multiple-path propagation), or to interfering sources which may be either deliberate or involuntary (in the case of co-channel transmissions). This is especially the case with radiocommunications in urban areas, subject to the phenomenon of multiple paths resulting from the reflections of the signal on surrounding fixed or moving obstacles potentially disturbed by the co-channel transmissions coming from the neighboring cells that re-use the same frequencies (in the case of F/TDMA or Frequency/Time Division Multiple Access networks). This is also the case with the HF (High Frequency) ionospherical links disturbed by the presence, at reception, of the multiple paths of propagation resulting from the reflections on the different ionospherical layers and of parasitic emitters due to high spectral congestion in the HF range.

For all these applications, whether it is for purposes of radiocommunications or for listening and technical analysis of the sources received, the sources need to be separated before other processing operations specific to the application considered are implemented. Furthermore, for certain applications such as passive listening, the sources received are totally unknown to the receiver (there are no available learning sequences, the waveforms are unknown, etc.) and their angular localization or goniometry may prove to be difficult (because of coupling between sensors) or costly (because of the calibration of the aerials) to implement. This is why it may prove to be highly advantageous to implement a source separation technique in a totally autodidactic or self-learning way, that is, by making use of no a priori information on the sources, apart from the assumption of the statistical independence of these sources.

The first studies on the separation of sources by self-learning appeared in the mid-1980s in the work of Jutten and Herault [1]. Since then, these studies have been constantly developing for mixtures of sources, both convolutive (time-spread multiple-path propagation channels) and instantaneous (distortion-free channels). A conspectus of these studies is presented in the article [2] by P. Comon and P. Chevalier. A certain number of techniques developed are called second-order techniques because they use only the information contained in the second-order statistics of the observations, as described in reference [3] for example. By contrast, other techniques, known as higher-order techniques, described for example in the reference [4], generally use not only second-order information but also information contained in statistics above the second order. These include the techniques known as cumulant-based, fourth-order techniques which have received special attention owing to their performance potential (reference [2]) and the relative simplicity of their implementation.

However, almost all the techniques of self-learned source separation available to date have been designed to separate sources assumed to be stationary, centered and ergodic, on the basis of estimators of statistics of observations qualified as being empirical, asymptotically unbiased and consistent on the basis of the above assumptions.

Two families of second-order separators are presently available. Those of the first family (F1) (reference [3], using the SOBI method shown schematically in FIG. 1) are aimed at separating statistically independent sources assumed to be stationary, centered and ergodic whereas those of the second family (F2) (reference [6], using the cyclic SOBI method) are designed to separate statistically independent sources assumed to be cyclostationary, centered and cycloergodic.

Two families of fourth-order separators are presently available. For example, those of the first family (F3) (reference [4] by J. F. Cardoso and A. Souloumiac, using the JADE method shown schematically in FIG. 2) are aimed at separating statistically independent sources assumed to be stationary, centered and ergodic while those of the second family (F4) (reference A. Ferreol and Chevalier [8] using the cyclic JADE technique) are designed to separate statistically independent sources assumed to be cyclostationary, centered and cycloergodic.

However, most of the sources encountered in practice are non-stationary and, more particularly, cyclostationary (with digitally modulated sources) and in certain cases deterministic (pure carriers). Furthermore, some of these sources are not centered. This is especially the case for deterministic sources and for certain digitally and non-linearly modulated sources as in the case of CPFSK sources with integer modulation index. This means that the empirical estimators of statistics classically used to implement the current techniques of self-learned source separation no longer have any reason to remain unbiased and consistent but are liable to become asymptotically biased. This may prevent the separation of the sources as shown in the document by A. Ferreol and P. Chevalier [5] for centered cyclostationary sources (linear digital modulations).

SUMMARY OF THE INVENTION

The invention relates to an antenna processing method for centered or potentially non-centered cyclostationary signals, comprising at least one step in which an nth order statistics estimator is obtained from r-order statistics, with r=1 to n−1, and for one or more values of r, a step for the correction of the empirical estimators, by means of r-order detected cyclic frequencies, exploiting the potentially non-centered character of the observations.

It comprises for example a step for the separation of the emitter sources of the signals received by using one of the second-order estimators or fourth-order estimators proposed.

It is also used for the angular localization or goniometry of the signals received.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the inventions shall appear more clearly from the following description of a non-exhaustive exemplary embodiment and from the appended figures, of which:

FIGS. 1 and 2 show prior art separation techniques, respectively known as the SOBI and JADE techniques,

FIG. 3 exemplifies a receiver according to the invention,

FIG. 4 is a diagram of the steps according to the invention applied to the SOBI technique of separation,

FIG. 5 is a diagram of the steps of the method according to the invention applied to the JADE technique of separation,

FIGS. 6A and 6B show results of separation using a classic estimator or an estimator according to the invention,

FIG. 7 shows the convergence of the estimator,

FIG. 8 is a drawing of the steps implemented according to the invention for the cyclic JADE method, and

FIG. 9 is a high level drawing of the steps implemented according to the invention.

MORE DETAILED DESCRIPTION

The method according to the invention enables the processing especially of potentially non-centered or centered cyclostationary sources. It can be applied, for example, to the separation of the sources or to their angular localization or goniometry. FIG. 9 illustrates wherein the method comprises at least the steps of: (802) receiving a mixture of cyclostationary and cycloergodic signals from independent sources; (804) generating an asymptotically unbiased and consistent estimator of a cyclic correlation matrix for centered observations of cyclostationary and cycloergodic signals; (806) generating an asymptotically unbiased and consistent estimator of a cyclic covariance matrix for non-centered observations of cyclostationary and cycloergodic signals; and (808) performing angular localization on the received signals using the at least one nth order estimator.

FIG. 3 gives a diagrammatic view of an exemplary receiver comprising, for example, N reception sensors (1 ₁, 1 ₂, 1 ₃, 1 ₄) each connected to several inputs of a processing device 2 such as a processor adapted to executing the steps of the method described here below.

Before describing various alternative modes of implementation of the method according to the invention, we shall describe the way to obtain the estimators, for example second-order or fourth-order estimators, as well as the detection of the cyclic frequencies.

Modelling of the Signal

It is assumed that an antenna with N sensors receives a noise-ridden mixture from P (P≦N) narrow-band (NB) and statistically independent sources. On the basis of these assumptions, the vector, x(t) of the complex envelopes of the output signals from the sensors is written, at the instant t, as follows:

$\begin{matrix} {{x(t)} = {{{\sum\limits_{p = 1}^{P}\;{{m_{p}(t)}{\mathbb{e}}^{j{({{2\pi\;\Delta\; f_{p}t} + \phi_{p}})}}a_{p}}} + {b(t)}}\overset{\Delta}{=}{{A\;{m_{c}(t)}} + {b(t)}}}} & (1) \end{matrix}$ where b(t) is the noise vector, assumed to be centered, stationary, circular and spatially white, m_(p)(t), Δf_(p), φ_(p) and a_(p) correspond respectively to the complex, narrow-band (NB), cyclostationary and potentially non-centered envelope (deterministic as the case may be), to the residue of the carrier, to the phase and to the directional vector of the source p, m_(c)(t) is the vector whose components are the signals m_(pc)(t) ^(Δ) m_(p)(t) exp[j(2πΔf_(p)t+φ_(p))] and A is the matrix (N×P) whose columns are the vectors a_(p). Statistics of the Observations First Order Statistics

In the general case of cyclostationary and non-centered sources, the first-order statistics of the vector x(t), given by (1), are written as follows:

$\begin{matrix} {{e_{x}(t)}\overset{\Delta}{=}{{E\left\lbrack {x(t)} \right\rbrack} = {{\sum\limits_{p = 1}^{P}{{e_{p}(t)}{\mathbb{e}}^{j{({{2\pi\;\Delta\; f_{p}t} + \phi_{p}})}}a_{p}}}\overset{\Delta}{=}{{\sum\limits_{p = 1}^{P}{{e_{pc}(t)}a_{p}}}\overset{\Delta}{=}{A\;{e_{m\; c}(t)}}}}}} & (2) \end{matrix}$ where e_(p)(t), e_(pc)(t) and e_(mc)(t) are the mathematical expectation values respectively of m_(p)(t), m_(pc)(t) and m_(c)(t). The vectors e_(p)(t) and e_(pc)(t) accept a Fourier series decomposition, and we obtain:

$\begin{matrix} {{e_{pc}(t)}\overset{\Delta}{=}{{E\left\lbrack {m_{pc}(t)} \right\rbrack} = {{\sum\limits_{\gamma_{pc} \in \Gamma_{pc}^{1}}{e_{pc}^{\gamma_{pc}}{\mathbb{e}}^{{j2\pi\gamma}_{pc}t}}} = {\sum\limits_{\gamma_{p} \in \Gamma_{p}^{1}}{e_{p}^{\gamma_{p}}{\mathbb{e}}^{j{\lbrack{{2{\pi(\;{{\Delta\; f_{p}} + \gamma_{p}})}t} + \phi_{p}}\rbrack}}}}}}} & (3) \end{matrix}$ where

Γ_(p)¹ = {γ_(p)}  and  Γ_(pc)¹ = {γ_(pc) = γ_(p) + Δ f_(p)}   are the sets of the cyclic frequencies γ_(p) and γ_(pc) respectively of e_(p)(t) and e_(pc)(t),

e_(p)^(γ_(p))  and  e_(pc)^(γ_(pc)) are the cyclic mean values respectively m_(p)(t) and m_(pc)(t), defined by

$\begin{matrix} {e_{p}^{\gamma_{p}}\overset{\Delta}{=}{< {{e_{p}(t)}{\mathbb{e}}^{{- {j2\pi\gamma}_{p}}t}} >_{c}}} & (4) \\ {e_{pc}^{\gamma_{pc}}\overset{\Delta}{=}{{< {{e_{pc}(t)}{\mathbb{e}}^{{- {j2\pi\gamma}_{pc}}t}} >_{c}} = {e_{p}^{\gamma_{{pc} - {\Delta\;{fp}}}}{\mathbb{e}}^{{j\phi}_{p}}}}} & (5) \end{matrix}$ where the symbol

${< {f(t)} >_{c}}\overset{\Delta}{=}\mspace{14mu}{\lim\limits_{T\rightarrow\infty}{\left( {1/T} \right){\int_{{- T}/2}^{T/2}{{f(t)}\ {\mathbb{d}t}}}}}$ corresponds to the operation of taking the temporal mean in continuous time f(t) on an infinite horizon of observation. Consequently, the vectors e_(mc)(t) and e_(x)(t) also accept a Fourier series decomposition and, by using (2) and (3), we get:

$\begin{matrix} {{e_{m\; c}(t)}\overset{\Delta}{=}{\sum\limits_{\gamma \in \Gamma_{x}^{1}}\;{e_{m\; c}^{\gamma}{\mathbb{e}}^{{j2\pi\gamma}\; t}}}} & (6) \\ {{e_{x}(t)}\overset{\Delta}{=}{{\sum\limits_{\gamma \in \Gamma_{x}^{1}}{e_{x}^{\gamma}{\mathbb{e}}^{{j2\pi\gamma}\; t}}} = {{\sum\limits_{\gamma \in \Gamma_{x}^{1}}{A\; e_{m\; c}^{\gamma}{\mathbb{e}}^{{j2\pi\gamma}\; t}}} = {\sum\limits_{p = 1}^{P}{\sum\limits_{\gamma_{pc} \in \Gamma_{pc}^{1}}{e_{pc}^{\gamma_{pc}}{\mathbb{e}}^{{j2\pi\gamma}_{pc}t}a_{p}}}}}}} & (7) \end{matrix}$ where

Γ_(x)¹ = ⋃_(1 ≤ p ≤ p){Γ_(pc)¹} is the set of the cyclic frequencies γ of e_(mc)(t) and e_(x)(t),

e_(m c)^(γ) and

e_(x)^(γ)  and  e_(x)^(γ) are respectively the cyclic means of m_(c)(t) and x(t), defined by:

$\begin{matrix} {e_{m\; c}^{\gamma} = {< {{e_{m\; c}(t)}{\mathbb{e}}^{- {j2\pi\gamma t}}} >_{c}}} & (8) \\ {e_{x}^{\gamma} = {< {{e_{x}(t)}{\mathbb{e}}^{- {j2\pi\gamma t}}} >_{c}}} & (9) \end{matrix}$

Assuming these conditions, the (quasi)-cyclostationary vector x(t) can be decomposed into the sum of a deterministic and (quasi)-periodic part e_(x)(t) and a (quasi)-cyclostationary, centered, random part Δx(t) such that: Δx(t) ^(Δ) x(t)−e _(x)(t)=AΔm _(c)(t)+b(t)  (10) where Δm_(c)(t) ^(Δ) m_(c)(t)−e_(mc)(t) is the centered vector of the source signals, with components Δm_(pc)(t) ^(Δ) m_(pc)(t)−e_(pc)(t)=Δm_(p)(t) e^(j(2πΔf) ^(p) ^(t+φ) ^(p) ⁾ where Δm_(p)(t) ^(Δ) m_(p)(t)−e_(p)(t). Special Cases

By way of an indication, e_(p)(t)=0 for a digitally and linearly modulated source p, which is centered. However, e_(p)(t)≠0 for a deterministic (carrier) source p as well as for certain digitally and non-linearly modulated sources such as CPFSKs with integer index, whose complex envelope is written as follows:

$\begin{matrix} {{m_{p}(t)} = {\pi_{p}^{1/2}{\sum\limits_{n}\;{\exp\left\{ {j\left\lbrack {\theta_{pn} + {2\pi\; f_{dp}{a_{n}^{p}\left( {t - {n\; T_{p}}} \right)}}} \right\rbrack} \right\}\mspace{14mu}{{Rect}_{p}\left( {t - {n\; T_{p}}} \right)}}}}} & (11) \end{matrix}$ where T_(p) corresponds to the symbol duration of the source, π_(p) ^(Δ) <E[|m_(p)(t)|²]>_(c) is the mean power of the source p received by an omnidirectional sensor, the

a_(n)^(p) are the transmitted M_(p)-ary symbols, assumed to be i.i.d and taking their values in the alphabet ±1, ±3, . . . , ±(M_(p)−1), where M_(p) is generally a power of 2, Rect_(p)(t) is the rectangular pulse with an amplitude 1 and a duration T_(p), f_(dp) ^(Δ) h_(p)/2T_(p) is the frequency deviation, h_(p) is the index of modulation of the source and θ_(pn), which corresponds to the accumulation of all the symbols up to the instant (n−1)T_(p), is defined by:

$\begin{matrix} {\theta_{pn}\overset{\Delta}{=}{2\pi\; f_{dp}T_{p}{\sum\limits_{k = {- \infty}}^{n - 1}\; a_{k}^{p}}}} & (12) \end{matrix}$ For M_(p)-ary symbols, the associated CPFSK source is called an M_(p)-CPFSK source. For this type of source, the set Γ_(p) ¹ of the first-order cyclic frequencies of the source p is written as

Γ_(p)¹ = {γ_(p) = ±(2k + 1)f_(dp), 0≦k≦(M_(p)−2)/2}. Thus we get:

$\begin{matrix} {e_{p}^{\gamma} = {{{\pm \pi_{p}^{1/2}}\frac{1}{M_{p}}\mspace{11mu}{for}\mspace{14mu}\gamma} \in \Gamma_{p}^{1}}} & (13) \end{matrix}$ Second-Order Statistics

Based on the above assumptions (non-centered, cyclostationary sources), the second-order statistics of the observations are characterized by the two correlation matrices R_(xε)(t, τ) for ε=1 and ε=−1, dependent on the current time t and defined by: R _(xε)(t,τ) ^(Δ) E[x(t)x(t−τ)^(εT) ]=AR _(mcε)(t,τ)A ^(εT)+η2(τ)δ(1+ε)I  (14) where ε=±1, with the convention x^(1Δ) x and x^(−1Δ) x*, * is the complex conjugation operation, δ(.) is the Kronecker symbol, ^(T) signifies transposed, η₂(τ) is the correlation function of the noise on each sensor, I is the identity matrix, the matrix R_(mcε)(t, τ) ^(Δ) E[m_(c)(t) m_(c)(t−τ)^(εT)] introduces the first and second correlation matrices of the vector m_(c)(t).

In the general case of non-centered cyclostationary sources using (10) in (14), the matrix R_(xε)(t, τ) takes the following form: R _(xε)(t,τ)=R _(Δxε)(t,τ)+e _(x)(t)e _(x)(t−τ)^(εT)  (15) where R_(Δxε)(t, τ) introduces the first and second matrices of covariance of x(t) or of correlation of Δx(t) , defined by R _(Δxε)(t,τ) ^(Δ) E[Δx(t)Δx(t−τ)^(εT) ]=AR _(Δmcε)(t, τ)A ^(εT)+η2(τ)δ(1+ε)I  (16) where R_(Δmcε)(t, τ) ^(Δ E[Δm) _(c)(t) Δm_(c)(t−τ)^(εT)] defines the first and second matrices of covariance of m_(c)(t) such that: R _(mcε)(t,τ)=R _(Δmcε)(t,τ)+e _(mc)(t)e _(mc)(t−τ)^(εT)  (17) Using (1), (10) and the assumption of statistical independence of the sources in (16), we get:

$\begin{matrix} \begin{matrix} {{R_{\Delta\;{xɛ}}\left( {t,\tau} \right)} = {{\sum\limits_{p = 1}^{P}\;{{r_{\Delta\; p\; ɛ}\left( {t,\tau} \right)}{\mathbb{e}}^{j{\lbrack{{{({1 + ɛ})}{({{2{\pi\Delta}\; f_{p}t} + \phi_{p}})}} - {2{\pi\Delta}\; f_{p}{\tau ɛ}}}\rbrack}}a_{p}a_{p}^{ɛ\; T}}} + {{\eta_{2}(\tau)}I}}} \\ {= {{\sum\limits_{p = 1}^{P}\;{{r_{\Delta\; p\;{cɛ}}\left( {t,\tau} \right)}a_{p}a_{p}^{ɛ\; T}}} + {{\eta_{2}(\tau)}{\delta\left( {1 + ɛ} \right)}I}}} \end{matrix} & (18) \end{matrix}$ where r_(Δpε)(t, τ) ^(Δ) E[Δm_(p)(t) Δm_(p)(t−τ)^(ε)], r_(Δpcε)(t, τ) ^(Δ) E[Δm_(pcε)(t) Δm_(pc)(t−τ)^(ε)]. From the expressions (2) and (18) a new writing of the expression (15) is finally deduced and is given by:

$\begin{matrix} {{R_{xɛ}\left( {t,\tau} \right)} = {{\sum\limits_{p = 1}^{P}{{r_{\Delta\; p\;{cɛ}}\left( {t,\tau} \right)}a_{p}a_{p}^{ɛ\; T}}} + {\sum\limits_{p = 1}^{P}{\sum\limits_{q = 1}^{P}{{e_{pc}(t)}{e_{qc}\left( {t - \tau} \right)}^{ɛ}a_{p}a_{p}^{ɛ\; T}}}} + {{\eta_{2}(\tau)}{\delta\left( {1 + ɛ} \right)}I}}} & (19) \end{matrix}$

For cyclostationary sources, the functions of covariance r_(Δpcε)(t, τ), 1≦p≦P, and hence the matrices R_(Δmcε)(t, τ) and R_(Δxε)(t, τ) are (quasi-) or polyperiodic functions of the time t accepting a Fourier series decomposition, and we get:

$\begin{matrix} {{r_{\Delta\; p\; c\; ɛ}\left( {t,\tau} \right)} = {\sum\limits_{\alpha_{\;^{{\Delta\; p\; c\; ɛ} \in {\Gamma\Delta}_{pc}^{\lbrack{1,ɛ}\rbrack}}}}{{r_{\Delta\;{pc}\; ɛ}^{\alpha_{\Delta\; p\; c\; ɛ}}(\tau)}{\mathbb{e}}^{{j2\pi}\;\alpha_{\Delta\;{pc}\; ɛ\; t}}}}} & (20) \\ {{R_{\Delta\; x\; ɛ}\left( {t,\tau} \right)} = {\sum\limits_{\alpha_{\;^{{\Delta\; x\; ɛ} \in {\Gamma\Delta}_{x}^{\lbrack{1,ɛ}\rbrack}}}}{{R_{\Delta\; x\; ɛ}^{\alpha_{\Delta\; x\; ɛ}}(\tau)}{\mathbb{e}}^{{j2\pi}\;\alpha_{\Delta\; x\; ɛ\; t}}}}} & (21) \end{matrix}$ where

Γ_(Δ pc)^([1, ɛ]) = {α_(Δ pc ɛ)} is the set of the cyclic frequencies α_(Δpcε)of

r_(Δ pc ɛ)(t, τ), Γ_(Δ x)^([1, ɛ]) = {α_(Δ x ɛ)} = ⋃_(1 ≤ p ≤ P){Γ_(Δ pc)^([1, ɛ])} is the set of cyclic frequencies α_(Δxε)of R_(Δxε)(t, τ), the quantities

r_(Δ pc ɛ)^(α_(Δp c ɛ))(τ) define the first (ε=−1) and second (ε=1) functions of cyclic covariance of m_(pc)(t) while the matrices

R_(Δ x ɛ)^(α_(Δ x ɛ))(τ) define the first and second matrices of cyclic covariance of x(t), respectively defined by

$\begin{matrix} {{r_{\Delta\;{pc}\; ɛ},^{\alpha_{\Delta\;{pc}\; ɛ}}{(\tau) = {< {{r_{\Delta\;{pc}\; ɛ}\left( {t,\tau} \right)}{\mathbb{e}}^{{- j}\; 2\pi\;\alpha_{\Delta\;{pc}\; ɛ}t}} >_{c}}}}\mspace{256mu}} & (22) \\ {R_{\Delta\; x\; ɛ},^{\alpha_{\Delta\; x\; ɛ}}{(\tau) = {{< {{R_{\Delta\; x\; ɛ}\left( {t,\tau} \right)}{\mathbb{e}}^{{- j}\; 2\pi\;\alpha_{\Delta\; x\; ɛ}t}} >_{c}} = {{A\mspace{11mu}{R_{\Delta\; m\; c\; ɛ}^{\alpha_{\Delta\; x\; ɛ}}(\tau)}A^{ɛ\; T}} + {{\eta_{2}(\tau)}{\delta\left( \alpha_{\Delta\; x\; ɛ} \right)}{\delta\left( {1 + ɛ} \right)}I}}}}} & (23) \end{matrix}$ where

R_(Δ m c ɛ)^(α_(Δ x ɛ))(τ) defines the first and second matrices of cyclic covariance of m_(c)(t). Using the expressions (7) and (21) in (15), we get

$\begin{matrix} {{R_{x\; ɛ}\left( {t,\tau} \right)} = {{\sum\limits_{\alpha_{\Delta\; x\; ɛ} \in \mspace{11mu}\Gamma_{\Delta_{x}}^{\lbrack{1,ɛ}\rbrack}}{{R_{\Delta\; x\; ɛ}^{\alpha_{\Delta\; x\; ɛ}}(\tau)}{\mathbb{e}}^{j\; 2\;{\pi\alpha}_{\Delta\; x\; ɛ}t}}} + {\sum\limits_{\gamma\; \in \;\Gamma_{x}^{1}}{\sum\limits_{\omega\; \in \;\Gamma_{x}^{1}}{e_{x}^{\gamma}e_{x}^{\omega\mspace{11mu} ɛ\mspace{11mu} T}{\mathbb{e}}^{{- j}\; 2\pi\;\omega\;\tau\; ɛ}{\mathbb{e}}^{j\; 2{\pi{({\gamma + {\omega\; ɛ}})}}t}}}}}} & (24) \end{matrix}$ which shows that the matrices R_(xε)(t, τ) accept a Fourier series decomposition

$\begin{matrix} {{R_{x\; ɛ}\left( {t,\tau} \right)} = {\sum\limits_{\alpha_{x\; ɛ} \in \mspace{11mu}\Gamma_{\;_{x}}^{\lbrack{1,ɛ}\rbrack}}{{R_{x\; ɛ}^{\alpha_{x\; ɛ}}(\tau)}{\mathbb{e}}^{j\; 2\;{\pi\alpha}_{x\; ɛ}t}}}} & (25) \end{matrix}$ where

Γ_(x)^([1, ɛ]) = {α_(x ɛ)} = Γ_(Δ x)^([1, ɛ])⋃{Γ_(x)¹o_(ɛ)Γ_(x)¹} is the set of the cyclic frequencies α_(xε) of

R_(x ɛ)(t, τ), Γ_(x)¹o_(ɛ)Γ_(x)¹ = {α = γ + ɛω  where  γ ∈ Γ_(x)¹, ω ∈ Γ_(x)¹}, the matrices

R_(x ɛ)^(α_(x ɛ))(τ) are the first (ε=−1) and second (ε=+1) matrices of cyclic correlation of x(t), defined by

$\begin{matrix} {{R_{x\; ɛ}^{\alpha_{x\; ɛ}}(\tau)} = {{< {{R_{x\; ɛ}\left( {t,\tau} \right)}{\mathbb{e}}^{{- j}\; 2\pi\;\alpha_{x\; ɛ}t}} >_{c}} = {{A\mspace{11mu}{R_{m\; c\; ɛ}^{\alpha_{x\; ɛ}}(\tau)}A^{ɛ\; T}} + {{\eta_{2}(\tau)}\mspace{11mu}{\delta\left( \alpha_{x\; ɛ} \right)}\mspace{11mu}{\delta\left( {1 + ɛ} \right)}\mspace{11mu} I}}}} & (26) \end{matrix}$ where

R_(mc ɛ)^(α_(x ɛ))(τ) defines the first and second matrices of cyclic correlation of m_(c)(t). In particular, for the zero cyclic frequency, the matrix

R_(x ɛ)^(α_(x ɛ))(τ) corresponds to the temporal mean in t, R_(xε)(τ), de R_(xε)(t, τ) which is written, using (14), R _(xε)(τ) ^(Δ) <R _(xε)(t,τ)>_(c) =AR _(mcε)(τ)A ^(εT)+η2(τ)δ(1+ε)I  (27) where, using (17), the matrix R_(mcε)(τ) ^(Δ) <R_(mcε)(t, τ)>_(c) is written R _(mcε)(τ) ^(Δ) <R _(mcε)(t,τ)>_(c) =R _(Δmcε)(τ)+<e _(mc)(t)e _(mc)(t−τ)^(εT)>_(c) =R _(Δmcε)(τ)+E _(mcε)(τ)  (28) where R_(Δmcε)(τ) ^(Δ) <R_(Δmcε)(t, τ)>_(c) and E_(mcε)(τ) ^(Δ) <e_(mc)(t) e_(mc)(t−τ)^(εT)>c. Empirical Estimator of the Second-Order Statistics

In practice, the second-order statistics of the observations are unknown in principle and must be estimated by the taking the temporal mean on a finite period of observation, on the basis of a number K of samples x(k) (1≦k≦K), of the observation vector x(t), using the property of ergodicity of these samples in the stationary case and of cycloergodicity of these samples in the cyclostationary case. If T_(e) denotes the sampling period, the estimation of the cyclic correlation matrix

R_(x ɛ)^(α)(τ) for τ=lT_(e), which corresponds to a matrix of cumulants only for centered observations, is done only by means of the estimator

R̂_(x ɛ)^(α)(lT_(e))(K) qualified as empirical and defined by

$\begin{matrix} {{{\hat{R}}_{x\; ɛ}^{\alpha}\left( {lT}_{e} \right)}(K)\mspace{14mu}\underset{\underset{\_}{\_}}{\Delta}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x(k)}\mspace{11mu}{x\left( {k - l} \right)}^{ɛ\; T}{\mathbb{e}}^{{- j}\; 2\pi\;\alpha\;{kT}_{e}}}}} & (29) \end{matrix}$

The method according to the invention comprises, for example, a novel step to determine a second-order estimator.

Novel Estimator of the Second-Order Cyclic Cumulants of the Observations

In the presence of non-centered observations, the matrices of correlation of the observations R_(xε)(t, τ) no longer correspond to the matrices of covariance or of second-order cumulants of the observations, as indicated by the expression (15). This is also the case with the matrices of cyclic correlation,

R_(x ɛ)^(α_(x ɛ))(τ), defined by (26) which, for non-centered observations, no longer correspond to the matrices of covariance or matrices of second-order cyclic cumulants

R_(Δ x ɛ)^(α_(x ɛ))(τ), defined by (22) for α_(Δxε)=α_(xε). Thus, an efficient operation of the second-order separators F1 and F2 with respect to the potentially non-centered cyclostationary sources can be obtained only by making use of the information contained in the matrices

R_(Δ x ɛ)^(α_(x ɛ))(τ) rather than in the matrices

R_(x ɛ)^(α_(x ɛ))(τ). Thus, in as much as, for cyclostationary and cycloergodic sources, the empirical estimator (29) is an asymptotically unbiased and consistent estimator of the matrix of cyclic correlation

R_(x ɛ)^(α)(lT_(e)), another estimator is used for non-centered observations. This other estimator is aimed at making an asymptotically unbiased and consistent estimation of the matrix of cyclic covariance

R_(Δ x ɛ)^(α)(lT_(e)).

From the expressions (24) and (25), we deduce the expression of the matrix of cyclic covariance

R_(Δ x ɛ)^(α)(lT_(e)) as a function of that of

R_(x ɛ)^(α)(lT_(e)), given by

$\begin{matrix} {{R_{\Delta\; x\; ɛ}^{\alpha}\left( {lT}_{e} \right)} = {{R_{x\; ɛ}^{\alpha}\left( {lT}_{e} \right)} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}{e_{x}^{\alpha - {\omega\; ɛ}}e_{x}^{\omega\; ɛ\; T}{\mathbb{e}}^{{- j}\; 2\pi\;\omega\; ɛ\;{lT}_{e}}}}}} & (30) \end{matrix}$

Thus, the estimation of the matrix of cyclic covariance

R_(Δ x ɛ)^(α)(lT_(e)) is made from the estimator

R̂_(Δ x ɛ)^(α)(lT_(e))(K) defined by

$\begin{matrix} {{{{\hat{R}}_{\Delta\; x\; ɛ}^{\alpha}\left( {lT}_{e} \right)}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}{{\hat{R}}_{x\; ɛ}^{\alpha}\left( {lT}_{e} \right)}(K)} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}{{{\hat{e}}_{x}^{\alpha - {\omega\; ɛ}}(K)}{{\hat{e}}_{x}^{\omega\;}(K)}^{ɛ\; T}{\mathbb{e}}^{{- j}\; 2\pi\;\omega\; ɛ\;{lT}_{e}}}}} & (31) \end{matrix}$ where

R̂_(Δ x ɛ)^(α)(lT_(e))(K) is defined by (29) and where

ê_(x)^(ω)(K) is defined by

$\begin{matrix} {{{\hat{e}}_{x}^{\omega}(K)}\overset{\Delta}{=}{\frac{1}{K}{\sum\limits_{k = 1}^{K}\;{{x(k)}{\mathbb{e}}^{{- j}\; 2\;\pi\;\omega\;{kT}_{e}}}}}} & (32) \end{matrix}$

Thus, assuming cyclostationary and cycloergodic observations, whether centered or not, the estimator (31) is an asymptotically unbiased and consistent estimator of the matrix of cyclic covariance or matrix of second-order cyclic cumulants

R_(Δ x ɛ)^(α)(l T_(e)). In particular, the separators F1 must exploit the estimator (31) for α=0 and ε=−1, written as

R̂_(Δ x)(l T_(e))(K), and defined by

$\begin{matrix} {{{{\hat{R}}_{\Delta\; x}\left( {l\; T_{e}} \right)}(K)}\overset{\Delta}{=}{{{{\hat{R}}_{\; x}\left( {l\; T_{e}} \right)}(K)} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}^{\;}\;{{{\hat{e}}_{x}^{\omega}(K)}{{\hat{e}}_{x}^{\omega}(K)}^{\dagger}{\mathbb{e}}^{j\; 2\;\pi\;\omega\; l\; T_{e}}}}}} & (33) \end{matrix}$ where {circumflex over (R)}_(x)(lT_(e))(K) is defined by (29) with α=0 and ε=−1.

According to another alternative embodiment, the method includes a step for determining a new fourth-order estimator.

Third-Order Statistics

Based on the above assumptions, the third-order statistics (moments) of the observations are defined by (34) T _(xε)(t,τ ₁,τ₂)[i,j,k] ^(Δ) E[x _(i)(t)x _(j)(t−τ ₁)^(ε) x _(k)(−τ ₂)]=E[Δx _(i)(t)Δx _(j)(t−τ ₁)^(ε) Δx _(k)(t−τ ₂)]+e _(x) [i](t)E[Δx _(j)(t−τ ₁)^(ε) Δx _(k)(t−τ ₂)]+e _(x) [k](t−τ ₂)E[Δx _(i)(t)Δx _(j)(t−τ ₁)^(ε) ]+e _(x) [j](t−τ ₁)^(ε) E[Δx _(i)(t)Δx _(k)(t−τ ₂)]+4e _(x) [i](t)e _(x) [j](t−τ ₁)^(ε) e _(x) [k](t−τ ₂) where ex_(x)[i](t)=e_(xi)(t) is the component i of e_(x)(t). The third-order cumulants are the quantities E[Δx_(i)(t)Δx_(j)(t−τ₁)^(ε)Δx_(k)(t−τ₂)]. Assuming that the term T_(xε)(t, τ₁, τ₂)[i, j, k] is the element [i, N(j−1)+k] of the matrix T_(xε)(t, τ₁, τ₂), with a dimension (N×N²), we get an expression of this matrix given by:

$\begin{matrix} {{T_{xɛ}\left( {t,\tau_{1},\tau_{2}} \right)} = {{A\;{T_{mce}\left( {t,\tau_{1},\tau_{2}} \right)}\left( {A \otimes A^{ɛ}} \right)^{ɛ\; T}} = {\sum\limits_{i,j,{k = 1}}^{P}\;{{{T_{mce}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}{a_{i}\left\lbrack {a_{j} \otimes a_{k}^{ɛ}} \right\rbrack}^{ɛT}}}}} & (35) \end{matrix}$ where T_(mcε)(t, τ₁, τ₂) is the matrix (P×P²) whose coefficients are the quantities T_(mcε)(t, τ₁, τ₂)[i, j, k] defined by

$\begin{matrix} {{{{{{T_{{mc}ɛ}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack} = {{E\left\lbrack {{m_{ic}(t)}{m_{jc}\left( {t - \tau_{1}} \right)}^{ɛ}{m_{{kc}\;}\left( {t - \tau_{2}} \right)}} \right\rbrack} = {{{E\left\lbrack {\Delta\;{m_{ic}(t)}{m_{ic}\left( {t - \tau_{1}} \right)}^{ɛ}\Delta\;{m_{ic}\left( {t - \tau_{2}} \right)}} \right\rbrack}\mspace{11mu}{\delta\left( {i - j} \right)}{\delta\left( {i - k} \right)}} + {{e_{ic}(t)}{E\left\lbrack {\Delta\;{m_{jc}\left( {t - \tau_{1}} \right)}^{ɛ}\Delta\;{m_{jc}\left( {t - \tau_{2}} \right)}} \right\rbrack}{\delta\left( {j - k} \right)}} + {{e_{kc}\left( {t - \tau_{2}} \right)}{E\left\lbrack {\Delta\;{m_{ic}(t)}\Delta\;{m_{ic}\left( {t - \tau_{1}} \right)}^{ɛ}} \right\rbrack}{\delta\left( {i - j} \right)}} +}}}\quad}{e_{jc}\left( {t - \tau_{1}} \right)}^{ɛ}{E\left\lbrack {\Delta\;{m_{ic}(t)}\;\Delta\;{m_{ic}\left( {t - \tau_{2}} \right)}} \right\rbrack}{\delta\left( {i - k} \right)}} + {4{e_{ic}(t)}{e_{jc}\left( {t - \tau_{1}} \right)}^{ɛ}{e_{kc}\left( {t - \tau_{2}} \right)}}} & (36) \end{matrix}$

For cyclostationary sources, the matrices of third-order moments, T_(mcε)(t, τ₁, τ₂) and T_(xε)(t, τ₁, τ₂), are (quasi-) or poly-periodic functions of the time t accepting a Fourier series decomposition and we get:

$\begin{matrix} {{T_{xɛ}\left( {t,\tau_{1},\tau_{2}} \right)} = {\sum\limits_{v_{xɛ} \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}\;{{T_{xɛ}^{v_{xɛ}}\left( {\tau_{1},\tau_{2}} \right)}{\mathbb{e}}^{{j2\pi}\; v_{xɛ}t}}}} & (37) \end{matrix}$ where

Γ_(x)^([1, ɛ1]) = {v_(xɛ)} is the set of the cyclic frequencies v_(xε)of T_(xε)(t, τ₁, τ₂) and

T_(xɛ)^(v_(xɛ))(τ₁, τ₂) a matrix of cyclic third-order moments of x(t), defined respectively by

$\begin{matrix} {{T_{xɛ}^{v_{xɛ}}\left( {\tau_{1},\tau_{2}} \right)} = {{< {{T_{xɛ}\left( {t,\tau_{1},\tau_{2}} \right)}{\mathbb{e}}^{{- {j2\pi}}\; v_{xɛ}t}} >_{c}} = {A\;{T_{mcɛ}^{v_{xɛ}}\left( {\tau_{1},\tau_{2}} \right)}\left( {A \otimes A^{ɛ}} \right)^{ɛ\; T}}}} & (38) \end{matrix}$ where {circle around (x)} corresponds to the Kronecker product. Fourth-Order Statistics

Based on the above assumptions (relating to non-centered and cyclostationary signals) the fourth-order statistics of the observations are the fourth-order cumulants defined by:

$\begin{matrix} {{{Q\; x\;{{\zeta\left( {t,\tau_{1},\tau_{2},\tau_{3}} \right)}\left\lbrack {i,j,k,l} \right\rbrack}}\overset{\Delta}{=}{{{Cum}\left( {{x_{i}(t)},{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1},{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2},{x_{l}\left( {t - \tau_{3}} \right)}} \right)} = {{{Cum}\left( {{{\Delta x}_{i}(t)},{\Delta\;{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}},{\Delta\;{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}},{\Delta\;{x_{l}\left( {t - \tau_{3}} \right)}}} \right)} = {{{E\left\lbrack {\Delta\;{x_{i}(t)}\Delta\;{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}\Delta\;{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}\Delta\;{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack} - {{E\left\lbrack {\Delta\;{x_{i}(t)}\Delta\;{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}} \right\rbrack}{E\left\lbrack {\Delta\;{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}\Delta\;{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{E\left\lbrack {\Delta\;{x_{i}(t)}\Delta\;{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}{E\left\lbrack {\Delta\;{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}\Delta\;{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{E\left\lbrack {\Delta\;{x_{i}(t)}\Delta\;{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}{E\left\lbrack {\Delta\;{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}\Delta\;{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}}} = {{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack} - {{e_{xi}(t)}{E\left\lbrack {{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{e_{xj}\left( {t - \tau_{1}} \right)}^{\zeta 1}{E\left\lbrack {{x_{i}(t)}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{e_{xk}\left( {t - \tau_{2}} \right)}^{\zeta 2}{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{e_{xl}\left( {t - \tau_{3}} \right)}{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}} - {{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}} \right\rbrack}{E\left\lbrack {{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{E\left\lbrack {{x_{i}(t)}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}{E\left\lbrack {{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} - {{E\left\lbrack {{x_{i}(t)}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}{E\left\lbrack {{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}} + {2{e_{xi}(t)}}}}}}}{{e_{xj}\left( {t - \tau_{1}} \right)}^{\zeta 1}{E\left\lbrack {{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} + {2{e_{xi}(t)}{e_{xk}\left( {t - \tau_{2}} \right)}^{\zeta 2}{E\left\lbrack {{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} + \;{2{e_{xi}(t)}{e_{xl}\left( {t - {\tau_{3}{E\left\lbrack {{{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}x_{k}} + \left( {t - \tau_{2}} \right)^{\zeta 2}} \right\rbrack}} + {2{e_{xk}\left( {t - \tau_{2}} \right)}^{\zeta 2}{e_{xj}\left( {t - \tau_{1}} \right)}^{\zeta 1}{E\left\lbrack {{x_{i}(t)}{x_{l}\left( {t - \tau_{3}} \right)}} \right\rbrack}} + {2{e_{xl}\left( {t - \tau_{3}} \right)}{e_{xj}\left( {t - \tau_{1}} \right)}^{\zeta 1}{E\left\lbrack {{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}{x_{i}(t)}} \right\rbrack}} + {2{e_{xk}\left( {t - \tau_{2}} \right)}^{\zeta 2}{e_{xl}\left( {t - \tau_{3}} \right)}{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}} \right\rbrack}} - {6\;{e_{xi}(t)}{e_{xj}\left( {t - \tau_{1}} \right)}^{\zeta 1}{e_{xk}\left( {t - \tau_{2}} \right)}^{\zeta 2}{e_{xl}\left( {t - \tau_{3}} \right)}}} \right.}}} & (39) \end{matrix}$ where ζ ^(Δ) (ζ₁, ζ₂) and (ζ₁, ζ₂)=(1, 1), (−1, 1) or (−1, −1). Assuming that the term Q_(xζ)(t, τ₁, τ₂, τ₃)[i, j, k, l] is the element [N(i−1)+j, N(k−1)+l] of the matrix Q_(xζ)(t, τ₁, τ₂, τ₃), known as the quadricovariance matrix, with a dimension (N²×N²), we obtain an expression of this dimension given by:

$\begin{matrix} {{{Q\;}_{x\;\zeta}\left( {t,\tau_{1},\tau_{2},\tau_{3}} \right)} = {{\left( {A \otimes A^{\zeta 1}} \right){Q_{m\; c\;\zeta}\left( {t,\tau_{1},\tau_{2},\tau_{3}} \right)}\left( {A^{\zeta 2} \otimes A} \right)^{T}} = {\sum\limits_{i,j,{k = 1}}\;{{Q_{mc\zeta}\left\lbrack {i,j,k,l} \right\rbrack}{{\left( {t,\tau_{1},\tau_{2},\tau_{3}} \right)\left\lbrack {a_{i.} \otimes a_{j}^{\zeta 1}} \right\rbrack}\left\lbrack {a_{k}^{\zeta 2} \otimes a_{l}} \right\rbrack}^{T}}}}} & (40) \end{matrix}$ where Q_(mcζ)(t, τ₁, τ₂, τ₃) is the quadricovariance of the vector m_(c)(t) whose elements are Q_(mcζ)[i, j, k, l](t, τ₁, τ₂, τ₃) ^(Δ) Cum(m_(ic)(t), m_(jc)(t−τ₁)^(ζ1), m_(kc)(t−τ₂)^(ζ2), m_(lc)(t−τ₃)).

For cyclostationary sources, the matrices of quadricovariance, Q_(mcζ)(t, τ₁, τ₂, τ₃) and Q_(xζ)(t, τ₁, τ₂, τ₃), are (quasi-) or poly-periodic functions of the time t accepting a Fourier series decomposition, and we obtain

$\begin{matrix} {{{Q\;}_{x\;\zeta}\left( {t,\tau_{1},\tau_{2},\tau_{3}} \right)} = {\sum\limits_{\beta_{x\zeta} \in \Gamma_{x}^{\lbrack{1,{\zeta 1},{\zeta 2},1}\rbrack}}{{Q\;}_{x\;\zeta}^{\beta_{x\zeta}}\left( {\tau_{1},\tau_{2},\tau_{3}} \right){\mathbb{e}}^{{j2\pi}\;\beta_{x\zeta}t}}}} & (41) \end{matrix}$ where

Γ_(x)^([1, ζ1, ζ2, 1]) = {β_(xζ)} is the set of cyclic frequencies β_(xζ)of Q_(xζ)(t, τ₁, τ₂, τ₃) and

Q_(xζ)^(β_(xζ))(τ ₁, τ ₂, τ ₃) is a matrix of cyclic quadricovariance of x(t), defined respectively by

$\begin{matrix} {{Q_{x\;\zeta}^{\beta_{x\;\zeta}}\left( {{\tau\;}_{1},{\tau\;}_{2},{\tau\;}_{3}} \right)} = {{< {{Q_{x\;\zeta}\left( {t,{\tau\;}_{1},{\tau\;}_{2},{\tau\;}_{3}} \right)}{\mathbb{e}}^{{- {j2\pi}}\;\beta_{x\;\zeta}t}} >_{c}} = {\left( {A \otimes A^{\zeta 1}} \right){Q_{m\; c}^{\beta_{x}}\left( {{\tau\;}_{1},{\tau\;}_{2},{\tau\;}_{3}} \right)}\left( {A^{\zeta 2} \otimes A} \right)^{T}}}} & (42) \end{matrix}$

In particular, the cyclic quadricovariance for the zero cyclic frequency corresponds to the temporal mean in t, Q_(xζ)(τ₁, τ₂, τ₃), de Q_(xζ)(t, τ₁, τ₂, τ₃), which is written as follows: Q _(xζ)(τ₁, τ₂, τ₃) ^(Δ) <Q _(xζ)(t, τ ₁, τ₂, τ₃)>_(c)=(A{circle around (x)}A ^(ζ1))Q _(mcζ)(τ₁, τ₂, τ₃)(A ^(ζ2) {circle around (x)}A)^(T)  (42b) where Q_(mcζ)(τ₁, τ₂, τ₃) is the temporal mean in t of Q_(mcζ)(t, τ₁, τ₂, τ₃). Novel Estimator of the Fourth-Order Cumulants of the Observations

From (39) and from the Fourier series decomposition of the statistics appearing in this expression, we deduce the expression of the element [i, j, k, l ] of the matrix of cyclic quadricovariance

Q_(xζ)^(β)(τ ₁, τ ₂, τ ₃) in the general case of non-centered observations, given by:

$\begin{matrix} {{{{{M_{x\zeta}^{\beta}\left( {{\tau\;}_{1},{\tau\;}_{2},{\tau\;}_{3}} \right)}\left\lbrack {i,j,k,l} \right\rbrack} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}\;\left\{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{{T_{x\zeta}^{\beta - \gamma}\left( {{{\tau\;}_{1} - {\tau\;}_{3}},{\tau_{2} - {\tau\;}_{3}}} \right)}\left\lbrack {l,j,k} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}{({\beta - \gamma})}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack j\rbrack}^{\zeta 1}{{T_{x\zeta 2}^{\beta - {\gamma\zeta 1}}\left( {\tau_{2},\tau_{3}} \right)}\left\lbrack {i,k,l} \right\rbrack}{\mathbb{e}}^{- {j2\pi\gamma\zeta 1\tau}_{1}}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{\zeta 2}{{T_{x\zeta 1}^{\beta - {\gamma\zeta 2}}\left( {\tau_{1},\tau_{3}} \right)}\left\lbrack {i,j,l} \right\rbrack}{\mathbb{e}}^{- {j2\pi\gamma\zeta 2\tau}_{2}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{{T_{x\zeta}^{\beta - \gamma}\left( {\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}{\mathbb{e}}^{- {j2\pi\gamma\tau}_{3}}}} \right\}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}\;{{{R_{x\zeta 1}^{\alpha}\left( \tau_{1} \right)}\left\lbrack {i,j} \right\rbrack}{{R_{x\zeta 2}^{{\zeta 2}{({\beta - \alpha})}}\left( {\tau_{3} - \tau_{2}} \right)}\left\lbrack {k,l} \right\rbrack}^{\zeta 2}{\mathbb{e}}^{{{j2\pi}{({\alpha - \beta})}}\tau_{2}}}} - {\sum\limits_{\gamma \in \Gamma_{x}^{\lbrack{1,{ɛ2}}\rbrack}}\;{{{R_{x\zeta 2}^{\gamma}\left( \tau_{2} \right)}\left\lbrack {i,k} \right\rbrack}{{R_{x\zeta 1}^{{\zeta 1}{({\beta - \gamma})}}\left( {\tau_{3} - \tau_{1}} \right)}\left\lbrack {j,l} \right\rbrack}^{\zeta 1}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}\tau_{1}}}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}\;{{{R_{x\zeta 1}^{\omega}\left( \tau_{3} \right)}\left\lbrack {i,l} \right\rbrack}{{R_{x\zeta 12}^{{\zeta 1}{({\beta - \omega})}}\left( {\tau_{2} - \tau_{1}} \right)}\left\lbrack {j,k} \right\rbrack}^{\zeta 1}{\mathbb{e}}^{{{j2\pi}{({\omega - \beta})}}\tau_{1}}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\;}\left\{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x\zeta 2}^{\beta - \gamma - {\omega\zeta 1}}\left( {\tau_{2} - \tau_{3}} \right)}\left\lbrack {l,k} \right\rbrack}{\mathbb{e}}^{{j2\pi\omega\zeta 1}{({\tau_{3} - \tau_{1}})}}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{\zeta 2}{{R_{x\zeta 1}^{\beta - \gamma - {\omega\zeta 2}}\left( {\tau_{1} - \tau_{3}} \right)}\left\lbrack {l,j} \right\rbrack}{\mathbb{e}}^{{j2\pi\omega\zeta 2}{({\tau_{3} - \tau_{2}})}}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack l\rbrack}{{R_{x\zeta 12}^{{\zeta 1}{({\beta - \gamma - \omega})}}\left( {\tau_{2} - \tau_{1}} \right)}\left\lbrack {j,k} \right\rbrack}^{\zeta 1}{\mathbb{e}}^{{j2\pi\omega}{({\tau_{1} - \tau_{3}})}}{\mathbb{e}}^{- {j2\pi\beta\tau}_{1}}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{\zeta 2}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x1}^{\beta - {\gamma\zeta 2} - {\omega\zeta 1}}\left( \tau_{3} \right)}\left\lbrack {i,l} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\gamma\tau}_{2}}{\zeta 2}}{\mathbb{e}}^{{- {j2\pi\omega\tau}_{1}}{\zeta 1}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x\;{\zeta 2}}^{\beta - \gamma - {\omega\zeta 1}}\left( \tau_{2} \right)}\left\lbrack {i,k} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\omega\tau}_{1}}{\zeta 1}}{\mathbb{e}}^{- {j2\pi\gamma\tau}_{3}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{\zeta 2}{{R_{x\;{\zeta 1}}^{\beta - \gamma - {\omega\zeta 2}}\left( \tau_{1} \right)}\left\lbrack {i,j} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\omega\tau}_{2}}{\zeta 2}}{\mathbb{e}}^{- {j2\pi\gamma\tau}_{3}}}} \right\}}}} - {6{\sum\limits_{\underset{{\mathbb{e}}^{{j2\pi\tau}_{3}{({\gamma - \beta})}}}{\gamma \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\;}{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{e_{x}^{\delta}\lbrack k\rbrack}^{\zeta 2}{e_{x}^{\beta - {\omega\zeta 1} - {\delta\zeta 2}}\lbrack l\rbrack}{\mathbb{e}}^{{j2\pi\omega\zeta 1}{({\tau_{3} - \tau_{1}})}}{\mathbb{e}}^{{j2\pi\delta\zeta 2}{({\tau_{3} - \tau_{2}})}}}}}}}}\;\;{{{{where}\mspace{14mu}{{T_{x\zeta}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}}\overset{\Delta}{=}{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{\zeta 1}{x_{k}\left( {t - \tau_{2}} \right)}^{\zeta 2}} \right\rbrack}},{{{T_{x\zeta}^{\beta}\left( {\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}\overset{\Delta}{=}{< {{{T_{x\zeta}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\beta}}\; t}} >_{c}}},{{{T_{xɛ}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}\overset{\Delta}{=}{E\left\lbrack {{x_{i}(t)}{x_{j}\left( {t - \tau_{1}} \right)}^{ɛ}{x_{k}\left( {t - \tau_{2}} \right)}} \right\rbrack}},\mspace{250mu}{{{T_{xɛ}^{\beta}\left( {\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack} = {< {{{T_{xɛ}\left( {t,\tau_{1},\tau_{2}} \right)}\left\lbrack {i,j,k} \right\rbrack}\;{\mathbb{e}}^{{- {j2\pi\beta}}\; t}} >_{c}}}}}\mspace{20mu}} & (43) \end{matrix}$

In particular, the matrix of cyclic quadricovariance exploited by the separators of the family F3 corresponds to the matrix Q_(xζ) ^(β)(τ₁, τ₂, τ₃) for β=0ζ=(−1, −1) and (τ₁, τ₂, τ₃)=(0, 0, 0) and its element Q_(xζ) ^(β)(τ₁, τ₂, τ₃)[i, j, k, l], denoted as Q_(x)[i, j, k, l], is written:

$\begin{matrix} {{{Q_{x}\left\lbrack {i,j,k,l} \right\rbrack}\overset{\Delta}{=}{{< {{Q_{x}\left( {t,0,0,0} \right)}\left\lbrack {i,j,k,l} \right\rbrack} >_{c}} = {{M_{x}^{0}\left\lbrack {i,j,k,l} \right\rbrack} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}\;\left\{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{T_{x}^{\gamma}\left\lbrack {j,l,k} \right\rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{T_{x}^{\gamma}\left\lbrack {j,i,k} \right\rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack j\rbrack}^{*}{T_{x}^{\gamma}\left\lbrack {i,k,l} \right\rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{*}{T_{x}^{\gamma}\left\lbrack {i,j,l} \right\rbrack}}} \right\}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{- 1}}\rbrack}}\left\{ {{{R_{x}^{\alpha}\left\lbrack {i,j} \right\rbrack}{R_{x}^{- \alpha}\left\lbrack {l,k} \right\rbrack}} + {{R_{x}^{\alpha}\left\lbrack {i,k} \right\rbrack}{R_{x}^{- \alpha}\left\lbrack {l,j} \right\rbrack}}} \right\}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}\;{{C_{x}^{\omega}\left\lbrack {i,l} \right\rbrack}{C_{x}^{\omega}\left\lbrack {j,k} \right\rbrack}^{*}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}{\underset{{\omega \in \Gamma_{x}^{1}}\;}{\sum\{}{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{R_{x}^{\omega - \gamma}\left\lbrack {l,k} \right\rbrack}}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{*}{R_{x}^{\omega - \gamma}\left\lbrack {l,j} \right\rbrack}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack l\rbrack}{C_{x}^{\omega + \gamma}\left\lbrack {k,j} \right\rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{*}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{C_{x}^{\omega + \gamma}\left\lbrack {i,l} \right\rbrack}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{R_{x}^{\omega - \gamma}\left\lbrack {i,k} \right\rbrack}} + {e_{x}^{\gamma}\left\lbrack {l\; 1{e_{x}^{\omega}\lbrack k\rbrack}^{*}{R_{x}^{\omega - \gamma}\left\lbrack {i,j} \right\rbrack}} \right\}} - {6{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\;}{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{e_{x}^{\delta}\lbrack k\rbrack}^{*}{e_{x}^{\delta + \omega - \gamma}\lbrack l\rbrack}}}}}}}}}{{{{where}\mspace{14mu}{M_{x}^{0}\left\lbrack {i,j,k,l} \right\rbrack}}\overset{\Delta}{=}{< {E\left\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}{x_{k}(t)}^{*}{x_{l}(t)}} \right\rbrack} >_{c}}},{{T_{x}^{\beta}\left\lbrack {i,j,k} \right\rbrack}\overset{\Delta}{=}{< {{E\left\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}{x_{k}(t)}} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\beta}}\; t}} >_{c}}},{{R_{x}^{\alpha}\left\lbrack {i,j} \right\rbrack}\overset{\Delta}{=}{< {{E\left\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\alpha}}\; t}} >_{c}}},{{C_{x}^{\alpha}\left\lbrack {i,j} \right\rbrack}\overset{\Delta}{=}{< {{E\left\lbrack {{x_{i}(t)}{x_{j}(t)}} \right\rbrack}{\mathbb{e}}^{{- {j2\pi\alpha}}\; t}} >_{c}.}}}} & (44) \end{matrix}$

Thus, the estimation of the matrix of cyclic quadricovariance

Q_(x ζ)^(β)(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l] is made on the basis of the estimator

Q̂_(x)^(β)ζ(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l](K) defined by:

$\begin{matrix} {{{{{\hat{Q}}_{x_{\zeta}}^{\beta}\left( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} \right)}\left\lbrack {i,j,k,l} \right\rbrack}(K)} = {{{{{\hat{M}}_{x_{\zeta}}^{\beta}\left( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} \right)}\left\lbrack {i,j,k,l} \right\rbrack}(K)} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}\left\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{{\hat{T}}_{x_{\zeta}}^{\beta - \gamma}\left( {{\left( {l_{1} - l_{3}} \right)T_{e}},{\left( {l_{2} - l_{3}} \right)T_{e}}} \right)}\left\lbrack {l,j,k} \right\rbrack}(K){\mathbb{e}}^{{- {{j2\pi}{({\beta\; - \gamma})}}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{T}}_{x_{\zeta 2}}^{\beta - {\gamma\zeta 1}}\left( {{l_{2}T_{e}},{l_{3}T_{e}}} \right)}\left\lbrack {i,k,l} \right\rbrack}(K){\mathbb{e}}^{{- {j2\pi\gamma\zeta 2l}_{1}}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{\zeta 2}{{{\hat{T}}_{x_{\zeta 2}}^{\beta - {\gamma\zeta 2}}\left( {{l_{1}T_{e}},{l_{3}T_{e}}} \right)}\left\lbrack {i,j,l} \right\rbrack}(K){\mathbb{e}}^{{- {j2\pi\gamma\zeta}_{2}}l_{2}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{{\hat{T}}_{x_{\zeta}}^{\beta - \gamma}\left( {{l_{1}T_{e}},{l_{2}T_{e}}} \right)}\left\lbrack {i,j,k} \right\rbrack}(K){\mathbb{e}}^{{- {j2\pi\gamma}}\; l_{3}T_{e}}}} \right\}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}{{{{\hat{R}}_{x_{\zeta 1}}^{\alpha}\left( {l_{1}T_{e}} \right)}\left\lbrack {i,j} \right\rbrack}(K){{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 2}{({\beta - \alpha})}}\left( {\left( {l_{3} - l_{2}} \right)T_{e}} \right)}\left\lbrack {k,l} \right\rbrack}(K)^{\zeta 2}{\mathbb{e}}^{{{j2\pi}{({\alpha - \beta})}}l_{2}T_{e}}}} - {\sum\limits_{\gamma \in \Gamma_{x}^{\lbrack{1,{ɛ2}}\rbrack}}{{{{\hat{R}}_{x_{\zeta 2}}^{\gamma}\left( {l_{2}T_{e}} \right)}\left\lbrack {i,k} \right\rbrack}(K){{{\hat{R}}_{x_{\zeta 1}}^{{\zeta 1}{({\beta - \gamma})}}\left( {\left( {l_{3} - l_{1}} \right)T_{e}} \right)}\left\lbrack {j,l} \right\rbrack}(K)^{\zeta 1}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}l_{1}T_{e}}}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}{{{{\hat{R}}_{x_{\zeta 1}}^{\omega}\left( {l_{3}T_{e}} \right)}\left\lbrack {i,l} \right\rbrack}(K){{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 1}{({\beta - \omega})}}\left( {\left( {l_{2} - l_{1}} \right)T_{e}} \right)}\left\lbrack {j,k} \right\rbrack}(K)^{\zeta 1}{\mathbb{e}}^{{{j2\pi}{({\omega - \beta})}}l_{1}T_{e}}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\;}\left\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{\zeta 2}}^{\beta - \gamma - \omega^{\zeta 1}}\left( {\left( {l_{2} - l_{3}} \right)T_{e}} \right)}\left\lbrack {l,k} \right\rbrack}(K){\mathbb{e}}^{{{j2\pi\omega\zeta}_{1}{({l_{3} - l_{1}})}}T_{e}}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{\zeta 2}{\hat{R}}_{x_{\zeta 1}}^{\beta - \gamma - \omega^{\zeta 2}}{\left( {\left( {l_{1} - l_{3}} \right)T_{e}} \right)\left\lbrack {l,j} \right\rbrack}(K){\mathbb{e}}^{{{j2\pi\omega\zeta 2}{({l_{3} - l_{2}})}}T_{e}}{\mathbb{e}}^{{{j2\pi}{({\gamma - \beta})}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack l\rbrack}(K){{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 1}{({\beta - \omega - \gamma})}}\left( {\left( {l_{2} - l_{1}} \right)T_{e}} \right)}\left\lbrack {j,k} \right\rbrack}(K)^{\zeta 1}{\mathbb{e}}^{{{j2\pi\omega}{({l_{1} - l_{3}})}}T_{e}}{\mathbb{e}}^{{- {j2\pi\beta}}\; l_{1}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{\zeta 2}{{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{1}}^{\beta - {\gamma\zeta 2} - {\omega\zeta 1}}\left( {l_{3}T_{e}} \right)}\left\lbrack {i,l} \right\rbrack}(K){\mathbb{e}}^{\;^{{- {j2\pi\gamma}}\; l_{2}T_{e}{\zeta 2}}}{\mathbb{e}}^{{- {j2\pi\omega}}\; l_{1}T_{e}{\zeta 1}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{\zeta 2}}^{\beta - \gamma - {\omega\zeta 1}}\left( {l_{2}T_{e}} \right)}\left\lbrack {i,k} \right\rbrack}(K){\mathbb{e}}^{\;^{{- {j2\pi\gamma}}\; l_{1}T_{e}{\zeta 1}}}{\mathbb{e}}^{{- {j2\pi\gamma}}\; l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{\zeta 2}{{{\hat{R}}_{x_{\zeta 1}}^{\beta - \gamma - {\omega\zeta 2}}\left( {l_{1}T_{e}} \right)}\left\lbrack {i,j} \right\rbrack}(K){\mathbb{e}}^{\;^{{- {j2\pi\omega}}\; l_{2}T_{e}{\zeta 2}}}{\mathbb{e}}^{{- {j2\pi\gamma}}\; l_{3}T_{e}}}} \right\}}}} - {6{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\;}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\;}{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{\hat{e}}_{x}^{\delta}\lbrack k\rbrack}(K)^{\zeta 2}{{\hat{e}}_{x}^{\beta - \gamma - {\omega\zeta 1} - {\delta\zeta 2}}\lbrack l\rbrack}(K){\mathbb{e}}^{{{j2\pi\omega\zeta 1}{({l_{3} - l_{1}})}}T_{e}}{\mathbb{e}}^{{{j2\pi\delta\zeta 2}{({l_{3} - l_{2}})}}T_{e}}{\mathbb{e}}^{{j2\pi}\; l_{3}{T_{e}{({\gamma - \beta})}}}}}}}}}} & (45) \end{matrix}$ where

M̂_(x)^(β)ζ(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l](K) is defined by the expression:

$\begin{matrix} {{{{{\hat{M}}_{x\mspace{11mu}\zeta}^{\beta}\left( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} \right)}\left\lbrack {i,j,k,l} \right\rbrack}(K)} = {\frac{1}{K}{\sum\limits_{m = 1}^{K}{{x_{i}(m)}{x_{j}\left( {m - l_{1}} \right)}{{{}_{}^{\zeta 1}{}_{}^{}}\left( {m - l_{2}} \right)}{{{}_{}^{\zeta 2}{}_{}^{}}\left( {m - l_{3}} \right)}{\mathbb{e}}^{{- j}\; 2\pi\;\beta\mspace{11mu} m\; T_{e}}}}}} & (46) \end{matrix}$ given in the reference [5] and where

ê_(x)^(γ)[i](K), R̂_(x ɛ)^(α)(lT_(e))[i, j](K), T̂_(x ɛ)^(α)(l₁T_(e), l₂T_(e))[i, j, l](K)  and  T̂_(x ζ)^(α)(l₁T_(e), l₂T_(e))[i, j, l](K)   are defined respectively by (47), (48), (49) and (50).

$\begin{matrix} {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{\mathbb{e}}^{{- {j2\pi}}\;\gamma\; k\; T_{e}}}}} & (47) \\ {{{{\hat{R}}_{x\; ɛ}^{\alpha}\left( {lT}_{e} \right)}\left\lbrack {i,j} \right\rbrack}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}\left( {k - l} \right)}{{}_{}^{}{}_{}^{{- j}\; 2\pi\;\alpha\; k\; T_{e}}}}}} & (48) \\ {{{{\hat{T}}_{x\; ɛ}^{\alpha}\left( {{l_{1}T_{e}},{l_{2}T_{e}}} \right)}\left\lbrack {i,j,l} \right\rbrack}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\;\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}\left( {k - l_{1}} \right)}{{\,^{ɛ}\; x_{l}}\left( {k - l_{2}} \right)}{\mathbb{e}}^{{- j}\; 2\pi\;\alpha\;{kT}_{e}}}}} & (49) \\ {{{{\hat{T}}_{x\;\zeta}^{\alpha}\left( {{l_{1}T_{e}},{l_{2}T_{e}}} \right)}\left\lbrack {i,j,l} \right\rbrack}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\;\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}\left( {k - l_{1}} \right)}{{{}_{}^{\zeta 1}{}_{}^{}}\left( {k - l_{2}} \right)}{{}_{}^{\zeta 2}{}_{}^{{- j}\; 2\pi\;\alpha\; k\; T_{e}}}}}} & (50) \end{matrix}$

Thus, assuming cyclostationary and cycloergodic observations, centered or non-centered, the estimator (45) is an asymptotically unbiased and consistent estimator of the element

Q_(x  ζ)^(β)(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l]. In particular, the separators F3 must exploit the estimator (45) for β=0, ζ=(−1, −1) and l₁=l₂=l₃=0, denoted {circumflex over (Q)}_(x)[i, j, k, l](K) and defined by:

$\begin{matrix} {{{{\hat{Q}}_{x}\left\lbrack {i,j,k,l} \right\rbrack}(K)\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}{{\hat{M}}_{x}^{0}\left\lbrack {i,j,k,l} \right\rbrack}(K)} - {\sum\limits_{\gamma\; \in \;\Gamma_{x}^{1}}\left\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{T}}_{x}^{\gamma}\left\lbrack {j,l,k} \right\rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{T}}_{x}^{\gamma}\left\lbrack {j,i,k} \right\rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack j\rbrack}(K)^{*}{{\hat{T}}_{x}^{\gamma}\left\lbrack {i,k,l} \right\rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{*}{{\hat{T}}_{x}^{\gamma}\left\lbrack {i,j,l} \right\rbrack}(K)}} \right\}} - {\sum\limits_{\omega\; \in \;\Gamma_{x}^{\lbrack{1,1}\rbrack}}{{{\hat{C}}_{x}^{\omega}\left\lbrack {i,l} \right\rbrack}(K){{\hat{C}}_{x}^{\omega}\left\lbrack {j,k} \right\rbrack}(K)^{*}}} - {\sum\limits_{\alpha\; \in \;\Gamma_{x}^{\lbrack{1,{- 1}}\rbrack}}\left\{ {{{{\hat{R}}_{x}^{\alpha}\left\lbrack {i,j} \right\rbrack}(K){{\hat{R}}_{x}^{- \alpha}\left\lbrack {l,k} \right\rbrack}(K)} + {{{\hat{R}}_{x}^{\alpha}\left\lbrack {i,k} \right\rbrack}(K){{\hat{R}}_{x}^{- \alpha}\left\lbrack {l,j} \right\rbrack}(K)}} \right\}} + {2{\sum\limits_{\gamma\; \in \;\Gamma_{x}^{1}}{\sum\limits_{\omega\; \in \;\Gamma_{x}^{1}}\left\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\left\lbrack {l,k} \right\rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\left\lbrack {l,j} \right\rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack l\rbrack}(K){{\hat{C}}_{x}^{\omega + \gamma}\left\lbrack {k,j} \right\rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{*}{{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{C}}_{x}^{\omega + \gamma}\left\lbrack {i,l} \right\rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\left\lbrack {i,k} \right\rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\left\lbrack {i,j} \right\rbrack}(K)}} \right\}}}} - {6{\sum\limits_{\gamma\; \in \;\Gamma_{x}^{1}}{\sum\limits_{\omega\; \in \;\Gamma_{x}^{1}}{\sum\limits_{\delta\; \in \;\Gamma_{x}^{1}}{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{e}}_{x}^{\delta}\lbrack k\rbrack}(K)^{*}{{\hat{e}}_{x}^{\delta + \omega - \gamma}\lbrack l\rbrack}(K)}}}}}} & (51) \end{matrix}$ where {circumflex over (M)}_(x) ⁰[i, j, k, l](K) is defined by (46) (defined here above) with β=0, l₁=l₂=l₃=0, ζ=(−1, −1), {circumflex over (T)}_(x) ^(γ)[i, j, l](K) is defined by (49) with α=γ, l₁=l₂=0, ε=−1, {circumflex over (R)}_(x) ^(α)[i, j] and Ĉ_(x) ^(α)[i, j] are defined by (29) with l=0 and respectively ε=−1 and ε=+1.

As indicated here above and as can be seen from the previous estimators, the method according the invention also comprises a step in which the second-order estimator is corrected by using cyclic frequencies which have to be detected a priori. The following detailed example is given in the case of a detection of the first-order cyclic frequencies.

Detector of the First-Order Cyclic Frequencies

The detection of the first-order cyclic frequencies of the observations γ by a detector of cyclic frequencies and the constitution of an estimate,

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ are done, for example, by computing the following standardized criterion:

$\begin{matrix} {{{V(\alpha)} = \frac{\frac{1}{N}{\sum\limits_{n = 1}^{N}{{{{\hat{e}}_{x}^{\alpha}\lbrack n\rbrack}\lbrack K\rbrack}}^{2}}}{{\hat{\gamma}}_{x}}}\mspace{445mu}{{With}\mspace{14mu}{\hat{\gamma}}_{x}} = {{\frac{1}{N}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{{x_{n}(k)}}^{2}\mspace{14mu}{and}\mspace{14mu}{{\hat{e}}_{x}^{\gamma}\lbrack n\rbrack}(K)}}} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{n}(k)}\;{\exp\left( {{- j}\; 2\;\pi\;\gamma\; k\; T_{e}} \right)}}}}}} & (52) \end{matrix}$ It may be recalled that x_(n)(t) is the signal received at the n^(th) sensor. The estimator ê_(x) ^(γ)[n](K) may be computed in an optimized way for K cyclic frequencies α_(k)=k/(K T_(e)) (0≦k≦K−1) by an FFT (a Fast Fourier Transformation) on the temporal samples x_(n)((k+k₀)T_(e)) such that 0≦k≦K−1. The criterion V(α) is standardized between 0 and 1 because γ _(x) represents the temporal mean of the mean power of the signal on the set of sensors. It being known that, on the assumption that x_(n)(kT_(e)) is a Gaussian noise, the criterion V(α) approximately follows a chi-2 relationship, the following detection test is deduced therefrom:

-   -   The frequency α is a cyclic frequency γ_(n) of E[x(t)]:         V(α)>=μ(pfα)     -   The frequency α is not a cyclic frequency: V(α)<μ(pfα)         where μ(pfα) is a threshold as a function of the probability of         false alarm pfα

The rest of the description gives several alternative modes of implementation of the method for the separation of statistically independent, stationary or cyclostationary sources. The associated separators are respectively called F′1, F′2, F′3 and F′4.

Proposed Second-Order Separators

Separators F1′

The separators of the family F1′ are self-learning second-order separators implementing the following operations:

Whitening Step

-   -   The detection of the first-order cyclic frequencies of the         observations γ by any detector of cyclic frequencies and the         constitution of an estimate,

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ.

-   -   The estimation of the matrix R_(Δx)(0) by {circumflex over         (R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 on the basis of         a given number K of samples.     -   The detection of the number of sources P from the decomposition         of {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All         the non-deterministic sources are detected).     -   The computation of the whitening matrix of the observations,         {circumflex over (T)}, where {circumflex over (T)} ^(Δ)         {circumflex over (Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension         (P×N), where {circumflex over (Λ)}_(s) is the diagonal matrix         (P×P) of the P greatest eigen values of {circumflex over         (R)}_(Δx)(0)(K)−λmin I, λmin is the minimum eigen value of         {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of         associated eigen vectors. We write z(t) ^(Δ) {circumflex over         (T)}x(t). (The directional vectors of the non-deterministic         sources are orthonormalized).         Identification Step     -   Choice of Q values of non-zero delays, l_(q), (1≦q≦Q).     -   For each value, l_(q), of the delay, estimation of the matrix of         averaged second-order cumulants of the observations,         R_(Δx)(l_(q)T_(e)), by {circumflex over (R)}_(Δx)(l_(q)T_(e))(K)         defined by (33), (29) and (32)     -   Computation of the matrices {circumflex over         (R)}_(Δz)(l_(q)T_(e))(K)         ^(Δ{circumflex over (T)}{circumflex over (R)})         _(Δx)(l_(q)T_(e))(K){circumflex over (T)}^(†)and self-learned         identification of the directional vectors of the whitened         sources by maximization, with respect to U ^(Δ) (u₁, u₂, . . .         up), of the criterion

${{C1}(U)}\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}{\sum\limits_{q = 1}^{Q}{\sum\limits_{l = 1}^{P}{{u_{l}^{\dagger}{R_{z}\left( \tau_{q} \right)}u_{l}}}^{2}}}$ given in the reference [7] where R_(z)(τ_(q)) is replaced by {circumflex over (R)}_(Δz)(l_(q)T_(e))(K). The solution matrix U is denoted Â_(nd)′ and contains an estimate of the whitened directional vectors of the non-deterministic sources. Filter

-   -   The computation of an estimate of the matrix of the directional         vectors of the non-deterministic source Â_(nd)=Û_(s){circumflex         over (Λ)}_(s) ^(1/2)Â_(nd)′     -   The extraction of the non-deterministic sources by any spatial         filtering of the observations constructed on the basis of         Â_(nd).         Processing of the Deterministic Sources     -   The construction of the orthogonal projector on the space         orthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)         ^(†)Â_(nd)]⁻¹Â_(nd) ^(†)     -   The implementation of the SOBI algorithm [3] on the basis of the         observations v(t) ^(Δ) Proj x(t) to identify the directional         vectors of the deterministic sources and extract them.         Separators F2′

The separators of the family F2′ are second-order self-learning separators implementing the following operations:

Whitening Step

-   -   The detection of the first order cyclic frequencies of the         observations γ by any unspecified detector of cyclic frequencies         and the constitution of an estimate,

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ.

-   -   The estimation of the matrix R_(Δx)(0) by {circumflex over         (R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 from a given         number of samples K     -   The detection of the number of sources P from the decomposition         into eigen elements of {circumflex over (R)}_(Δx)(0)(K). (all         the non-deterministic sources are detected).         Identification Step     -   The computation of the whitening matrix of the observations,         {circumflex over (T)}, where {circumflex over (T)} ^(Δ)         {circumflex over (Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension         (P×N), where {circumflex over (Λ)}_(s) is the diagonal matrix         (P×P) of the P biggest eigen values of {circumflex over         (R)}_(Δx)(0)(K)−λmin I, λmin is the minimal eigen value of         {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of the         associated eigen vectors. We write z(t) ^(Δ) {circumflex over         (T)}x(t). (The directional vectors of the non-deterministic         sources are orthonormalized).     -   The detection of the second-order cyclic frequencies of the         observations [9], α_(ε), for ε=−1 and ε=+1 by any detector of         cyclic frequencies and constitution of the estimates,

Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]), the sets respectively, Γ_(x) ^([1, −1]) and Γ_(x) ^([1, 1]), the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

-   -   The choice of an arbitrary number of pairs (α_(m), ε_(m))         (1≦m≦M) such that, for each of these pairs, at least one source         possesses the second-order cyclic frequency α_(m) for a matrix         of correlation indexed by ε_(m).     -   For each pair (α_(m), ε_(m)), (1≦m≦M):         -   The choice of an arbitrary number Q_(m) of delays, l_(mq),             (1≦q≦Q_(m))             -   For each value of the delay, l_(mq) (1≦q≦Q_(m)),             -   The estimation of the matrix of second-order cyclic                 cumulants of the whitened observations,

R_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) defined by (31), (29), (32) with the index z instead of x.

-   -   -   The detection of the number of non-deterministic sources             P_((αm, εm)) having the second-order cyclic frequency α_(m),             for the matrix of correlation indexed by ε_(m). This             detection test may consist of a search for the maximum rank             of the signal space of the matrices         -   {circumflex over (R)}_(Δzε), ^(αm) _(m)(l_(mq)T_(e))(K).             Û_((αm, εm)) denotes the unit matrix (P×P_((α) _(m, εm))),             obtained by SVD of the previous matrices, whose columns             generate the space generated by the whitened directional             vectors associated with the sources having the second-order             cyclic frequency, α_(m), for the matrix of correlation             indexed by ε_(m).         -   Reduction of dimension: We write v(t) ^(Δ) Û_((αm, εm))             ^(†)z(t), with a dimension (P_((αm, εm))×1) and carry out a             computation, for each delay l_(mq) (1≦q≦Q_(m)), of the             matrices

R̂_(Δ v ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) = Û_((α m, ɛ m))^(†)R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)Û_((α m, ɛ m))^(*ɛ_(m))

-   -   -   The self-learned identification of the doubly whitened             directional vectors associated with the pair (α_(m), ε_(m))             by maximization in relation to U ^(Δ) (u₁, u₂, . . . ,             up_((αm, εm))), of the criterion

${{C1}(U)}\mspace{11mu}\underset{\underset{\_}{\_}}{\Delta}\mspace{11mu}{\sum\limits_{q = 1}^{Q}{\sum\limits_{l = 1}^{P}{{u_{l}^{\dagger}{R_{z}\left( \tau_{q} \right)}u_{l}}}^{2}}}$ given in the reference [7] where R_(z)(τ_(q)) is replaced by

R̂_(Δ v ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)R̂_(Δ v ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)^(†). The solution matrix U is written as Â_(nd(αm, εm))′ and contains an estimate of the doubly whitened directional vectors of the non-deterministic sources associated with the pair (α_(m), ε_(m)). Filter

-   -   The computation of an estimate of the matrix of the directional         vectors of the non-deterministic sources associated with the         pair (α_(m), ε_(m)): Â_(nd(αm, εm))=Û_(s){circumflex over         (Λ)}_(s) ^(1/2)Û_((αm, εm))Â_(nd(αm, εm))′     -   The concatenation of the matrices Â_(nd(αm, εm)) for all the         pairs (α_(m), ε_(m)), (1≦m≦M). We obtain the matrix (N×P) of         Â_(nd) of the directional vectors of the non-deterministic         sources.     -   The extraction of the non-deterministic sources by any spatial         filtering of the observations constructed from Â_(nd).         Processing of the Deterministic Sources     -   The construction of the orthogonal projector on the space         orthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)         ^(†)Â_(nd)]⁻¹Â_(nd) ^(†)     -   The implementation of the SOBI algorithm [3] from the         observations w(t) ^(Δ) Proj x(t) to identify the directional         vectors of the deterministic sources and extract them.         Proposed Fourth-Order Separators         Separators F3′

The separators of the family F3′ are fourth-order self-learning separators implementing the following operations:

Whitening Step

-   -   The detection of the first-order cyclic frequencies of the         observations γ by any detector of cyclic frequencies and the         constitution of an estimate,

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ.

-   -   The detection of the second-order cyclic frequencies of the         observations, α_(ε), for ε=−1 and ε=+1 by any detector of cyclic         frequencies and the constitution of the estimates,

Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]), of the sets respectively,

-   -   Γ_(x) ^([1, −1])and

Γ̂_(x)^([1, 1]), of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

-   -   The estimation of the matrix R_(Δx)(0) by {circumflex over         (R)}_(Δx)(0)(K) defined by (53) and (52) for l=0 from a given         number of samples K     -   The detection of the number of sources P from the decomposition         of {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All         the non-deterministic sources are detected).     -   The computation of the whitening matrix of the observations,         {circumflex over (T)}, where {circumflex over (T)} ^(Δ)         {circumflex over (Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension         (P×N), where {circumflex over (Λ)}_(s) is the diagonal matrix         (P×P) of the P greatest eigen values of {circumflex over         (R)}_(Δx)(0)(K)−λmin I, λmin is the minimum eigen value of         {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of         associated eigen vectors. We write z(t) ^(Δ) {circumflex over         (T)}x(t). (The directional vectors of the non-deterministic         sources are orthonormalized).         Identification Step     -   The estimation of the quadricovariance, Q_(z), of the vector         z(t) by the expressions (51), (29), (49), (2) and (32) with the         index z instead of x.     -   The decomposition into eigen elements of {circumflex over         (Q)}_(z) and the estimation of the P eigen matrices M_(zi)         (1≦i≦P) associated with the P eigen values of higher-value         moduli.     -   The joint diagonalization of the P eigen matrices M_(zi)         weighted by the associated eigen values and the obtaining of the         matrix of the whitened directional vectors of the         non-deterministic sources Â_(nd)′.         Filter     -   The computation of an estimate of the matrix of the directional         vectors of the non-deterministic sources Â_(nd)=Û_(s){circumflex         over (Λ)}_(s) ^(1/2)Â_(nd)′     -   The extraction of the non-deterministic sources by any spatial         filtering of the observations constructed from Â_(nd).         Processing of the Deterministic Sources     -   The construction of the orthogonal projector on the space         orthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)         ^(†)Â_(nd)]⁻¹Â_(nd) ^(†)     -   The implementation of the algorithm JADE [4] from the         observations v(t)ΔProj x(t) to identify the directional vectors         of the deterministic sources and extract them.         Separators F4′

The separators of the family F4′ are fourth-order self-learning separators associated with the reference [8] implementing the following operations:

Whitening Step

-   -   The detection of the first-order cyclic frequencies of the         observations γ by any detector of cyclic frequencies and the         constitution of an estimate

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ.

-   -   The detection of the second-order cyclic frequencies of the         observations, α_(ε), for ε=−1 and ε=+1 by any detector of cyclic         frequencies and the constitution of the estimates,

Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]), of the sets respectively,

Γ_(x)^([1, −1])  and  Γ_(x)^([1, 1]), of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

-   -   The estimation of the matrix R_(Δx)(0) by {circumflex over         (R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 from a given         number of samples K.     -   The detection of the number of sources P from the decomposition         of {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All         the non-deterministic sources are detected).     -   The computation of the whitening matrix of the observations,         {circumflex over (T)}, where {circumflex over (T)} ^(Δ)         {circumflex over (Λ)}_(s) ^(−1/2) Û_(s) ^(\), with a size (P×N),         where {circumflex over (Λ)}_(s) is the diagonal matrix (P×P) of         the P greatest eigen values of {circumflex over         (R)}_(Δx)(0)(K)−λmin I, λmin is the minimum eigen value of         {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of the         associated eigen vectors. We write z(t) ^(Δ) {circumflex over         (T)} x(t). (The directional vectors of the non-deterministic         sources are orthonormalized).         Identification Step     -   The choice of a triplet (α_(m), ε_(m), l_(mq))     -   The estimation of the matrix of second-order cyclic cumulants of         the whitened observations,

R_(Δ z ɛ_(m) )^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)   defined by (31), (29), (32).

-   -   The computation of a unit matrix Û_((αm,εm))=[e₁. . . e_(p),]         where (P_((αm, εm))≦P) corresponds to the number of non-zero         eigen values of

R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) and e_(k) (1≦k≦P′) are the eigen vectors of

R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) associated with the P_((αm, εm)) highest eigen values.

-   -   The choice of the set (α_(m), ζ_(m), l_(m1)T_(e), l_(m2)T_(e),         l_(m3)T_(e)).     -   The reduction of dimension: v(t) ^(ΔÛ) _((αm, εm)) ^(†)z(t) is         written, with a dimension (P_((αm, εm))×1) and a computation is         made of the estimate of the cyclic quadricovariance

Q̂_(v ζ m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K) of v(t) from

Q̂_(v ζ m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K) and from Û_((αm, ζm)).

-   -   The decomposition into eigen elements of

Q̂_(v ζ m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K) and the estimation of the P_((αm, εm)) eigen matrices M_(vi) (1≦i≦P_((αm, εm))) associated with the P_((αm, εm)) eigen values with higher-value moduli.

-   -   The joint diagonalization of the P_((αm, εm)) eigen matrices         M_(vi)M_(vi) ^(†)weighted by the associated eigen values and the         obtaining of the matrix of the directional vectors of the doubly         whitened non-deterministic sources associated with the set         (α_(m), ζ_(m), l_(m1), l_(m2), l₃): Â_(nd) _((αm, ζm))′         Filter     -   The computation of an estimate of the matrix of the directional         vectors of the non-deterministic sources associated with the         pair (α_(m), ζ_(m)): Â_(nd(αm, ζm))=Û_(s){circumflex over         (Λ)}_(s) ^(1/2)Û_((αm, εm))Â_(nd(αm, εm))′     -   The concatenation of the matrices Â_(nd(αm, ζm)) for all the         pairs (α_(m), ζ_(m)), (1≦m≦M). The matrix (N×P) Â_(nd) of the         directional vectors of the non-deterministic sources is         obtained.     -   The extraction of the non-deterministic sources by any spatial         filtering whatsoever of the observations constructed on the         basis of Â_(nd).         Processing of the Deterministic Sources     -   The construction of the orthogonal projector on the space         orthogonal to the columns of Â_(nd):

Â_(nd) : Proj = I − Â_(nd)[Â_(nd)^(†)Â_(nd)]⁻¹Â_(nd)^(†)

-   -   The implementation of the algorithm JADE [4] from the         observations w(t) ^(Δ) Proj x(t) to identify the directional         vectors of the deterministic sources and extract these sources.         Exemplary Simulation of the Method According to the Invention         with the Separator F3′

FIGS. 6A and 6B respectively give a view, in a level graph expressed in dB, for two non-centered cyclostationary sources with a direction (↓₁=50° and ↓₂=60°), of the spectrum level after separation of the sources (in using the method JADE bringing into play the set of first-order cyclic frequencies and the two sets of second-order cyclic frequencies) and of the channel formation, after separation of the sources by using the classic estimator (FIG. 6A) and by using the proposed estimator (FIG. 6B).

It can be seen that, with the empirical estimator, the separation works badly because a source is localized at ↓=55° while the sources have angles of incidence of ↓₁=50° and ↓₂=60°. However, with the proposed estimator, the two sources are localized at 50° and 60° because the channel formation method on the first identified vector finds a maximum at 49.8° and, on the second vector, it finds a maximum at 60.1°. The signal-to-noise+interference ratios at output of filtering of the two sources are summarized in the following table:

Source at θ₁ = 50° Source at θ₂ = 60° Empirical estimators SNIR₁ = 15.3 dB SNIR₂ = 7.35 dB R_(xa) and Q_(xa) Proposed estimators SNIR₁ = 22 dB SNIR₂ = 22 dB R_(x) and Q_(x)

In the following simulation, the proposed estimator is known as an exhaustive estimator. This example keeps the same signal configuration as earlier when the two sources nevertheless have a signal-to-noise ratio of 10 dB. The classic empirical estimator is therefore compared with the exhaustive estimator. The SNIR in dB of the first source is therefore plotted as a function of the number of samples K.

FIG. 7 shows that the exhaustive estimator converges on the asymptote of the optimum interference canceling filter when K→+∞ and that the classic estimator is biased in converging on an SNIR (signal-to-noise+interference ratio) of 5 dB at output of filtering.

Separators F4′

The separators of the family F4′ are fourth-order self-learning separators implementing the following operations:

-   -   The detection of the first-order cyclic frequencies of the         observations γ by any detector whatsoever of cyclic frequencies         and the constitution of an estimate

Γ̂_(x)¹, of the set,

Γ_(x)¹, of the cyclic frequencies γ. The estimation of

Γ̂_(x)¹, can be done as in F1′.

-   -   The detection of the second-order cyclic frequencies of the         observations, α_(ε), for ε=−1 and ε=+1 by any detector         whatsoever of cyclic frequencies and the constitution of the         estimates,

Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]), of the sets respectively,

Γ_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]), of the cyclic frequencies respectively of the first and second matrices of correlation of the observations.

-   -   The estimation of the matrix R_(Δx)(0) by {circumflex over         (R)}_(Δx)(0)(K) defined by (53) and (52) for l=0 from a given         number of samples K.     -   The detection of the number of sources P from the decomposition         of {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All         the non-deterministic sources are detected).     -   The computation of the whitening matrix of the observations,         {circumflex over (T)}, where {circumflex over (T)}         ^(Δ{circumflex over (Λ)}) _(s) ^(−1/2) Û_(s) ^(†), with a size         (P×N), where {circumflex over (Λ)}_(s) is the diagonal matrix         (P×P) of the P greatest eigen values of {circumflex over         (R)}_(Δx)(0)(K)−λmin I, λmin is the minimum eigen value of         {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of the         associated eigen vectors. We write z(t) ^(Δ) {circumflex over         (T)}x(t). (The directional vectors of the non-deterministic         sources are orthonormalized).     -   The choice of a triplet (α_(m), ε_(m), l_(mq))     -   The estimation of the matrix of second-order cyclic cumulants of         the whitened observations,

R_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ z ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) defined by (31), (29), (32).

-   -   The computation of a unit matrix

Û_((α m, ɛ m)) = [e₁  ⋯  e_(p^(′))]  where  (P_((α m , ɛ m)) ≤ P) corresponds to the number of non-zero eigen values of

  R̂_(Δ z ɛ)^(α_(m)),  _(m)(l_(mq)T_(e))(K) and e_(k) (1≦k≦P_((αm, εm))) are the eigen vectors of

R̂_(Δ z ɛ)^(α_(m)),  _(m)(l_(mq)T_(e))(K) associated with the P_((αm, εm))) higher eigen values.

-   -   The reduction of dimension: v(t) ^(Δ) Û_((αm, εm)) ^(†)z(t) is         written with a dimension (P_((αm, εm))×1) and a computation is         made of the estimate of the cyclic quadricovariance

Q̂_(v ζ m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)  of  v(t) from

Q̂_(v ζ m )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)  and  from  Û_((α m, ζ m)),

-   -   The choice of a set (α_(m), ζ_(m), l_(m1), l_(m2), l_(m3))         (1≦m≦M) such that, for each of these pairs, at least one source         possesses the fourth-order cyclic frequency α_(m) for a         quadricovariance indexed by ζ_(m) and (l_(m1)T_(e), l_(m2)T_(e),         l_(m3)T_(e)),     -   The set of these values is chosen, for example, so that there is         compatibility with the parameters of the second-order cumulants         (α_(m), ε_(m), l_(mq))     -   For each set (α_(m), ζ_(m), l_(m1), l_(m2), l_(m3)) (1≦m≦M)     -   The estimation of the cyclic quadricovariance,

Q̂_(v ζ m )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e)), of the vector v(t) . We obtain

Q̂_(v ζ m )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K).

-   -   The decomposition into eigen elements of

Q̂_(v ζ m )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K) and the estimation of the P_((αm, εm)) eigen matrices M_(vi)(1≦i≦P_((αm, εm))) associated with the P_((αm, εm)) eigen values with higher-value moduli.

-   -   The joint diagonalization of the P_((αm, εm)) eigen matrices         M_(vi)M_(vi) ^(†)weighted by the associated eigen values and the         obtaining of the matrix of the directional vectors of the doubly         whitened non-deterministic sources associated with the set         (α_(m), ζ_(m), l_(m1), l_(m2), l_(m3)): Â_(nd(αm, ζm))′     -   The computation of an estimate of the matrix of the directional         vectors of the non-deterministic sources associated with the         pair (α_(m), ζ_(m)): Â_(nd(αm, ζm))=Û_(s){circumflex over         (Λ)}_(s) ^(1/2)Û_((αm, εm))Â_(nd(αm, εm))′     -   The concatenation of the matrices Â_(nd(αm, ζm)) for all the         pairs (α_(m), ζ_(m)), (1≦m≦M). The matrix (N×P) Â_(nd) of the         directional vectors of the non-deterministic sources is         obtained.     -   The extraction of the non-deterministic sources by any spatial         filtering whatsoever of the observations constructed on the         basis of Â_(nd).     -   The construction of the orthogonal projector on the space         orthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)         ^(†)Â_(nd)]⁻¹Â_(nd) ^(†)     -   The implementation of the algorithm JADE [4] from the         observations w(t) ^(Δ) Proj x(t) to identify the directional         vectors of the deterministic sources and extract them.

The implementation is summarized in FIG. 8 for M=1 (only one set (α_(m), ζ_(m), l_(m1), l_(m2), l_(m3)) is used).

The steps of the method according to the invention described here above are applied especially in the above-mentioned separation techniques, namely the SOBI, cyclic SOBI, JADE and cyclic JADE techniques.

Without departing from the scope of the invention, the steps of the method described here above are used, for example, to carry out the angular localization or goniometry of signals received at the level of a receiver.

For classic angular localization, the method uses for example:

-   -   MUSIC type methods described in the reference [10] of R. O         Schmidt with the matrix of covariance: {circumflex over         (R)}_(Δx)(0)(K)     -   Cyclic goniometry type methods given in the reference [11]         by W. A. Gardner with the matrix of cyclic covariance: R_(Δxε),         ^(αm) _(m)(l_(mq)T_(e))     -   Fourth-order goniometry methods described in the reference [12]         by P. Chevalier, A. Ferreol, J P. Denis with the         quadricovariance: {circumflex over (Q)}_(xζ), ⁰ _(m)(0, 0, 0)(K)     -   Methods of fourth-order cyclic goniometry with the cyclic         quadricovariance

Q̂_(x ζ m )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

Bibliography

-   [1] C. JUTTEN, J. HERAULT, <<Blind separation of sources, Part I: An     adaptive algorithm based on neuromimetic architecture>>, Signal     Processing, Elsevier, Vol 24, pp 1-10, 1991. -   [2] P. COMON, P. CHEVALIER, <<Blind source separation: Models,     Concepts, Algorithms and Performance>>, Chapter 5 of the book     Unsupervised adaptive filtering—Tome 1—Blind Source Separation, pp.     191-235, Dir. S. Haykins, Wiley, 445 p, 2000. -   [3] A. BELOUCHRANI, K. ABED-MERAIM, J. F. CARDOSO, E. MOULINES, <<A     blind source separation technique using second-order statistics>>,     IEEE Tran. Signal Processing, Vol. 45, N^(o). 2, pp 434-444,     February 1997. -   [4] J. F. CARDOSO, A. SOULOUMIAC, <<Blind beamforming for     non-Gaussian signals>>, IEEE Proceedings-F, Vol. 140, N^(o)6, pp     362-370, December 1993. -   [5] A. FERREOL, P. CHEVALIER, <<On the behavior of current second     and higher order blind source separation methods for cyclostationary     sources>>, IEEE Trans. Signal Processing, Vol 48, N^(o)6, pp.     1712-1725, June 2000. -   [6] K. ABED-MERAIM, Y. XIANG, J. H. MANTON, Y. HUA, <<Blind source     separation using second-order cyclostationary statistics>>, IEEE     Trans. Signal Processing, Vol. 49, N^(o)4, pp 694-701, April 2001. -   [7] P. CHEVALIER, <<Optimal separation of independent narrow-band     sources—Concept and Performance>>, Signal Processing, Elsevier,     Special issue on Blind Source Separation and Multichannel     Deconvolution, Vol 73, N^(o)1-2, pp. 27-47, February 1999. -   [8] A. FERREOL, P. CHEVALIER, <<Higher Order Blind Source Separation     using the Cyclostationarity Property of the Signals>>, ICASSP,     Munich (Germany), pp 4061-4064, April 1997. -   [9] SV SCHELL and W. GARDNER, “Detection of the number of     cyclostationnary signals in unknown interference and noise”, Proc of     Asilonan conf on signal, systems and computers 5-7 November 90. -   [10] R. O Schmidt, A signal subspace approach to multiple emitters     location and spectral estimation, PhD Thesis, Stanford University,     CA, November 1981. -   [11] W. A. Gardner, “Simplification of MUSIC and ESPRIT by     exploitation cyclostationarity”, Proc. IEEE, Vol. 76 No. 7, July     1988. -   [12] P. Chevalier, A. Ferreol, J P. Denis, “New geometrical results     about 4-th order direction finding methods performances”, EUSIPCO,     Trieste, pp. 923-926, 1996. 

1. An antenna processing method that enables separation or angular localization of cyclostationary signals, comprising: receiving at an antenna with N sensors, a mixture of cyclostationary and cycloergodic signals from independent sources; making at least one nth order estimator including: generating an asymrtotically unbiased and consistent estimator of a cyclic correlation matrix for centered observations of cyclostationary and cycloergodic signals; and generating an asymrtotically unbiased and consistent estimator of a cyclic covariance matrix for non-centered observations of cyclostationary and cycloergodic signals; thereby allowing separation, or angular localization of the received signals using the at least one nth order estimator.
 2. The method according to claim 1, comprising a step for separating emitter sources of the signals received by using the at least one estimator.
 3. The method according to claim 2 wherein the at least one estimator is a second-order estimator.
 4. The method according to claim 2 wherein the at least one estimator is a fourth-order estimator.
 5. The method according to claim 2 wherein the cyclic frequencies are detected first-order or second-order frequencies.
 6. The method of separating non-centered cyclostationary sources, according to claim 1, further comprising separating the signals using at least one of SSOBI, cyclic SOBI, JADE, and cyclic JADE separation techniques. 